Number Theory Seminar

Speaker: Christelle Vincent.

Title: Computing equations of hyperelliptic curves whose Jacobian has CM.

Abstract: It is known that given a CM sextic field, there exists a non-empty finite set of abelian varieties of dimension 3 that have complex multiplication by this field. Under certain conditions on the field and the CM type, this abelian variety can be guaranteed to be principally polarizable and simple. This ensures that the abelian variety is the Jacobian of a hyperelliptic curve or a plane quartic curve.
In this talk, we begin by showing how to generate a full set of period matrices for each isomorphism class of simple, principally polarized abelian variety with CM by a sextic field K. We then show how to determine whether the abelian variety is a hyperelliptic or plane quartic curve. Finally, in the hyperelliptic case, we show how to compute Rosenhain invariants for the curve.
This is joint work with J. Balakrishnan, S. Ionica and K. Lauter.